What would happen to Earth's orbit if it increased in mass or lost velocity?

[GordonDengyo]

Would it begin to spiral into the sun or would it slip into a new orbit? Don't most meteorite impacts occur due to the Earth sweeping into the meteor like a car hitting bugs? Wouldn't each and every impact slow the Earth down a tiny bit while also adding to the mass of the planet? I understand that the effect would be minimal(like a bug hitting my windshield) but wouldn't the accumulation of these tiny blows and additions of mass have an effect over millions of years?

 

[Nabeshin]

Well in polar coordinates the equation of orbit is given by:

$$r(t)=\frac{h^{2}}{\mu(1+ecos(\theta))}$$
Definitions:
$$\vec{h}=\vec{r}\times\vec{v}$$

$$\mu=GM , M>>m$$ where m is the mass of the orbiting body, in this case the Earth.

Since mu is the only thing in the equation that could depend on the Earth's mass, the only reason for the orbit to change would be for the assumption to no longer be valid (The mass of the orbiting body becomes comparable to that of the body being orbited). Seeing as the mass of the Earth is on the order of 10^24 and the mass gain per year is on the order of 10^10 (being generous), it will not happen in the life time of the universe.

 

[Ich]

I understand that the effect would be minimal(like a bug hitting my windshield) but wouldn't the accumulation of these tiny blows and additions of mass have an effect over millions of years?

Yes, these impacts will drain kinetic energy from earth, and Earth will react by moving closer to the sun and - paradoxically - moving faster than before. To understand the latter effect, imagine that you made the Earth stop on its orbit. It will start falling into the sun and, doing so, become actually faster than it was before you made it stop. It gained potential energy from sun's gravitational field.